The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
quot(x, 0) → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0, 0, 0)
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0, s(v))

Rewrite Strategy: INNERMOST

(1) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(2) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
quot(x, 0') → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0', s(v))

S is empty.
Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
quot(x, 0') → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro → 0':s:quotZeroErro → true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro → 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if2 :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat → 0':s:quotZeroErro

(5) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
le, quotIter

They will be analysed ascendingly in the following order:
le < quotIter

(6) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
quot(x, 0') → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro → 0':s:quotZeroErro → true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro → 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if2 :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat → 0':s:quotZeroErro

Generator Equations:
gen_0':s:quotZeroErro3_0(0) ⇔ 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) ⇔ s(gen_0':s:quotZeroErro3_0(x))

The following defined symbols remain to be analysed:
le, quotIter

They will be analysed ascendingly in the following order:
le < quotIter

(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Induction Base:
le(gen_0':s:quotZeroErro3_0(0), gen_0':s:quotZeroErro3_0(0)) →_R^Ω(1)
true

Induction Step:
le(gen_0':s:quotZeroErro3_0(+(n5_0, 1)), gen_0':s:quotZeroErro3_0(+(n5_0, 1))) →_R^Ω(1)
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) →_IH
true

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(8) Complex Obligation (BEST)

(9) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
quot(x, 0') → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro → 0':s:quotZeroErro → true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro → 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if2 :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat → 0':s:quotZeroErro

Lemmas:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:quotZeroErro3_0(0) ⇔ 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) ⇔ s(gen_0':s:quotZeroErro3_0(x))

The following defined symbols remain to be analysed:
quotIter

(10) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol quotIter.

(11) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
quot(x, 0') → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro → 0':s:quotZeroErro → true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro → 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if2 :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat → 0':s:quotZeroErro

Lemmas:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:quotZeroErro3_0(0) ⇔ 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) ⇔ s(gen_0':s:quotZeroErro3_0(x))

No more defined symbols left to analyse.

(12) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(13) BOUNDS(n^1, INF)

(14) Obligation:

Innermost TRS:
Rules:
le(0', y) → true
le(s(x), 0') → false
le(s(x), s(y)) → le(x, y)
quot(x, 0') → quotZeroErro
quot(x, s(y)) → quotIter(x, s(y), 0', 0', 0')
quotIter(x, s(y), z, u, v) → if(le(x, z), x, s(y), z, u, v)
if(true, x, y, z, u, v) → v
if(false, x, y, z, u, v) → if2(le(y, s(u)), x, y, s(z), s(u), v)
if2(false, x, y, z, u, v) → quotIter(x, y, z, u, v)
if2(true, x, y, z, u, v) → quotIter(x, y, z, 0', s(v))

Types:
le :: 0':s:quotZeroErro → 0':s:quotZeroErro → true:false
0' :: 0':s:quotZeroErro
true :: true:false
s :: 0':s:quotZeroErro → 0':s:quotZeroErro
false :: true:false
quot :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
quotZeroErro :: 0':s:quotZeroErro
quotIter :: 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
if2 :: true:false → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro → 0':s:quotZeroErro
hole_true:false1_0 :: true:false
hole_0':s:quotZeroErro2_0 :: 0':s:quotZeroErro
gen_0':s:quotZeroErro3_0 :: Nat → 0':s:quotZeroErro

Lemmas:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

Generator Equations:
gen_0':s:quotZeroErro3_0(0) ⇔ 0'
gen_0':s:quotZeroErro3_0(+(x, 1)) ⇔ s(gen_0':s:quotZeroErro3_0(x))

No more defined symbols left to analyse.

(15) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
le(gen_0':s:quotZeroErro3_0(n5_0), gen_0':s:quotZeroErro3_0(n5_0)) → true, rt ∈ Ω(1 + n5₀)

(16) BOUNDS(n^1, INF)